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Math Help - What am I doing wrong with this proof?

  1. #1
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    What am I doing wrong with this proof?

    So, I'm trying to use induction to prove:


    \sum_{k=0}^n\left(\frac{-1}{3}\right)^k = \frac{3}{4} + \frac{1}{4}(-1)^n3^{-n},~~ ~~for~all~n \in \mathbb{N}

    Base case is true.

    Assume:

    \left(\frac{-1}{3}\right)^k = \frac{3}{4} + \frac{1}{4}(-1)^n3^{-n}

    Test:

    \left(\frac{-1}{3}\right)^{k+1} = \frac{3}{4} + \frac{1}{4}(-1)^{(n+1)}3^{-(n+1)}

    \left(\frac{-1}{3}\right)^k\left(\frac{-1}{3}\right) = \frac{3}{4} + \frac{1}{4}(-1)^{n}(-1)3^{-n}3^{-1}

    \left(\frac{3}{4} + \frac{1}{4}(-1)^n3^{-n}\right)\left(\frac{-1}{3}\right) = \frac{3}{4} -\frac{1}{12}(-1)^{n}3^{-n}

    This next step is what seems to cause the problems. If the 3/4 on the LHS was not in brackets, everything would be fine and become = when I multiply the other term by -1/3; however, by multiplying the 3/4 by -1/3 it makes them not equal:

    -\frac{1}{4} - \frac{1}{12}(-1)^n3^{-n} = \frac{3}{4} -\frac{1}{12}(-1)^{n}3^{-n}

    What am I doing wrong?
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  2. #2
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    Re: What am I doing wrong with this proof?

    What happened to all of the summation signs??

    You're trying to prove something about the entire summation \sum_{k=0}^n\left\(\frac{-1}{3}\right\)^k, not the individual term \left\(\frac{-1}{3}\right\)^k.
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  3. #3
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    Re: What am I doing wrong with this proof?

    Quote Originally Posted by terrorsquid View Post
    So, I'm trying to use induction to prove:
    \sum_{k=0}^n\left(\frac{-1}{3}\right)^k = \frac{3}{4} + \frac{1}{4}(-1)^n3^{-n},~~ ~~for~all~n \in \mathbb{N}
    Without induction.
    Let S_n  = \sum\limits_{k = 0}^n {r^k }

    \begin{align*}  S_n&= 1+r+r^2+\cdots+r^n \\  rS_n &=r+r^2+r^3\cdots+r^{n+1}\\(1-r)S_n &=1-r^{n+1}\\ S_n &= \frac{1-r^{n+1}}{1-r} \end{align*}
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  4. #4
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    Re: What am I doing wrong with this proof?

    Quote Originally Posted by topspin1617 View Post
    What happened to all of the summation signs??
    It was a magic sum sign

    haha, stayed up too late last night I think.

    Thanks.
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